Question

Great Plains Roofing and Siding Company Inc. sells roofing and siding products to home repair retailers, such as Lowe's and Home Depot, and commercial contractors. The owner is interested in studying the effects of several variables on the value of shingles sold ($000). The marketing manager is arguing that the company should spend more money on advertising, while a market researcher suggests it should focus more on making its brand and product more distinct from its competitors. The company has divided the United States into 26 marketing districts. In each district, it collected information on the following variables: volume of sales (in thousands of dollars), advertising dollars (on thousands), number of active accounts, number of competing brands, and a rating of district potential Sales Advertising Number of Number of Market (000s) Dollars (000s) Accounts Competitors Potential 79.3 5.5 10 8.0 163.2 67 3.0 146.0 3.0 77.7 2.9 30.9 8.0 291.9 60.0 339.4 59.6 9.0 4.0 42 6.5 5.5 5.0 73 16 86.3 2375 6.0 5.0 3.5 8.0 107.2 55.0 10 10 291.4 14 6.0 135.8 223.3 7.0 6.7 6.1 3.6 4.2 195.0 73.4 477 140.7 13 93.5 259.0 331.2 26 75 5.6 71 A real estate developer wishes to study the relationship between the size of home a client will purchase (in square feet) and other variables. Possible independent variables include the family income, family size, whether there is a senior adult parent living with the family (1 for yes, 0 for no), and the total years of education beyond high school for the husband and wife. The sample information is reported below Square Income Family Senior Family Feet (000s) Size Parent Education 60.8 2,380 68.4 3,640 104.5 3,360 89.3 3,080 2,940 72.2 10 4,480 1254 83.6 133 2,520 4.200 10 2,800 95 a. Develop an appropriate mutiple regression equation using stopwise method. (Use Excel data analysis and enter number of family members first, then their income and delete any insignificant variables. Leave no cells blank be certain to enter "0" wherever required. Round your answers to 2 decimal places.) Step Family T-Value P-Value b. Select all independent variables that should be in the final model. (Select all that apply.) Senior parent Square feet Family size Income Education

Answer 1

1)

First, the data is entered in Excel:

Sales	Ad_Dollars	Accounts	Competitors	Potential
79.3	5.5	31	10	8
200.1	2.5	55	8	6
163.2	8	67	12	9
200.1	3	50	7	16
146	3	38	8	15
177.7	2.9	71	12	17
30.9	8	30	12	8
291.9	9	56	5	10
160	4	42	8	4
339.4	6.5	73	5	16
159.6	5.5	60	11	7
86.3	5	44	12	12
237.5	6	50	6	6
107.2	5	39	10	4
155	3.5	55	10	4
291.4	8	70	6	14
100.2	6	40	11	6
135.8	4	50	11	8
223.3	7.5	62	9	13
195	7	59	9	11
73.4	6.7	53	13	5
47.7	6.1	38	13	10
140.7	3.6	43	9	17
93.5	4.2	26	8	3
259	4.5	75	8	19
331.2	5.6	71	4	9

The data is first visualized through a matrix of scatterplots:

4 6 8 12 o O o O Sales og oogo Oo Ad _Dollars 2 o 0 o O O Accounts | |oo o o 0 O OC 0 0 O0 0 O O Competitors oO O o O 0 O o O Potential e OO 0 50 200 350 30 50 70 5 10 15

The above diagram shows that the Sales is weakly correlated with Ad Dollars and mildly with Potential, positively correlated with Accounts, and negatively correlated with Competitors. We first build a linear model as function of all variables:

> tt <- read.csv("clipboard",header=TRUE,sep="\t")
> pairs(tt)
> sales_lm <- lm(Sales~.,tt)
> summary(sales_lm)

Call:
lm(formula = Sales ~ ., data = tt)

Residuals:
     Min       1Q   Median       3Q      Max
-19.0906 -5.9796   0.8968   6.5667 14.7985

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) 178.3203    12.9603 13.759 5.62e-12 ***
Ad_Dollars    1.8071     1.0810   1.672    0.109   
Accounts      3.3178     0.1629 20.368 2.60e-15 ***
Competitors -21.1850     0.7879 -26.887 < 2e-16 ***
Potential     0.3245     0.4678   0.694    0.495   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.604 on 21 degrees of freedom
Multiple R-squared: 0.9892,    Adjusted R-squared: 0.9871
F-statistic: 479.1 on 4 and 21 DF, p-value: < 2.2e-16

The model is significant, and the covariates Accounts and Potential are significant variables.

The overall elasticity is 40.68815.

> predict(sales_lm,newdata=list(Ad_Dollars=8,Accounts=30,Competitors=12,Potential=8)
+ )
       1
40.68815

2)

a)

family	square feet	income	family size	senior parent	parent education
1	2240	60.8	2	0	4
2	2380	68.4	2	1	6
3	3640	104.5	3	0	7
4	3360	89.3	1	1	0
5	3080	72.2	4	0	2
6	2940	114.3	1	1	10
7	4480	125.4	6	0	6
8	2520	83.6	3	0	8
9	4200	133.5	5	0	2
10	2800	95.3	3	0	6

Regression Analysis: square feet versus income family ....nt, Education

Stepwise selection of terms

α to enter = 0.15, α to remove = 0.15

Analysis of Variance

Source DF Adj SS Adj MS F-value P-value

Regression 2 4619699 2309849 30.90 0.000

income 1 380692 380692 5.09 0.059

family size 1 962595 962595 12.88 0.009

Error 7 523341 74763

Total 9 5143040

Model Summary

S R-sq R-sq(adj) R-sq(pred)

273.428 89.82% 86.92% 82.48%

Coefficients

Term Coef SE Coef T-value p-value VIF

Constant 713 363 1.96 0.091

income 12.15 5.38 2.26 0.0059 2.08

family size 372 104 3.59 0.009 2.08

Regression Equation

Square feet = 713 + 12.15 income + 372 family size

Fits and Diagnostics for Unusual Observations

Obs Square feet fit Resid Std Resid   

3 3640 3098 542 2.25 R

R large residual

From the above output, the multiple linear regression equation is,

Y = 713 + 12.15 (income, X1) + 372 (Family size, X2)

b)

Family size and income are independent variables. That should be in the final model.